The present invention relates to dynamic systems, and more particularly to non-integer order dynamic systems such as, for example, proportional, integral, and derivative (PID) controllers.
The idea of fractional integro-differential operators (i.e., derivatives and integrals) seems to be a somewhat unusual topic, and is very hard to explain due to the fact that, unlike commonly used differential operators, it is not related to some direct geometrical meaning, such as the trend of functions or their convexity.
A number N of first order differential equations usually model complex dynamics. In system theory, we call N the degree of the system. Moreover, the theory of Laplace transforms in linear systems provides the possibility of analyzing the input-output relationship of a system via the ratio of its Laplace transform, which is called the transfer function of the system and whose denominator is a polynomial of degree N.
For that reason, such a mathematical tool could be judged far from reality. But many physical phenomena have an intrinsic fractional order description, and fractional order calculus is necessary to explain them. In fact, some physical phenomena exist whose analysis involves transfer functions of degree m, with m being a non-integer number.
Transmission lines, various kinds of electrical noises, dielectric polarization and heat transfer phenomena are some of the fields involving non-integer order physical laws. Evidence of this is provided in the following articles: J. C. Wang, xe2x80x9cRealization Of Generalized Warburg Impedance With RC Ladder And Transmission Linesxe2x80x9d, J. Electrochem. Soc., vol. 134, No. 8, pp. 1915-1940, 1987; M. S. Keshner, xe2x80x9c1/f Noisexe2x80x9d, Proceedings of the IEEE, vol. 70, No. 3, pp. 212-218, March 1982; B. Mandelbrot, xe2x80x9cSome Noises With 1/f Spectrum, A Bridge Between Direct Current And White Noisexe2x80x9d, IEEE Trans. Inform. Theory, vol. IT-13, No. 2, pp. 289-298, 1967; P. Arena, R. Caponetto, L. Fortuna and D. Porto, xe2x80x9cNonlinear Norder Circuits And Systems: An Introductionxe2x80x9d, Nonlinear Science, Series A, Vol. 38. World Scientific; A. Le Mehaute"", xe2x80x9cFractal Geometriesxe2x80x9d, CRC Press Inc. Boca Ratonxe2x80x94Ann Arborxe2x80x94London, 1991; and B. Onaral, H. P. Schwan, xe2x80x9cLinear And Non-linear Properties Of Platinum Electrode Polarization, Part I, Frequency Dependence At Very Low Frequenciesxe2x80x9d Met. Bio. Eng. Comput., vol. 20, pp. 299-306, 1982.
Following, for instance, K. B. Oldham, J. Spanier, xe2x80x9cFractional Calculusxe2x80x9d, Academic Press, N.Y., 1974 or B. Ross, Editor, xe2x80x9cFractional Calculus And Its Applicationsxe2x80x9d, Berlin: Springer Verlag, 1975, the most common definition of a non-integer order integral is the following:                                                         ⅆ                              -                q                                      ⁢                          f              ⁡                              (                t                )                                                          ⅆ                          t                              -                q                                                    =                              1                          Γ              ⁡                              (                q                )                                              ⁢                                    ∫              0              t                        ⁢                                                            (                                      t                    -                    τ                                    )                                                  q                  -                  1                                            ⁢                              f                ⁡                                  (                  τ                  )                                            ⁢                              ⅆ                τ                                                                        (        1        )            
The lower limit is chosen to be zero (it could be any real number) because in the following, time series with t greater than 0 will be considered. In the above definition, xcex93(q) is the factorial function defined for a positive real value q, by the following expression:                               Γ          ⁡                      (            q            )                          =                              ∫            0            ∞                    ⁢                                    x                              q                -                1                                      ⁢                          e                              -                x                                      ⁢                          xe2x80x83                        ⁢                          ⅆ              x                                                          (        2        )            
for which, when q is an integer, it holds that:
xcex93(q+1)=q!xe2x80x83xe2x80x83(3)
The definition of fractional derivative easily derives from (1) by taking an n order derivative (n being a suitable integer) of an m order integral (m suitable non-integer) to obtain an nxe2x88x92m=qxe2x88x92order one:                                                         ⅆ              q                        ⁢                          f              ⁡                              (                t                )                                                          ⅆ                          t              q                                      =                                                            ⅆ                                  n                  -                  m                                            ⁢                              f                ⁡                                  (                  t                  )                                                                    ⅆ                              t                                  n                  -                  m                                                              =                                                                      ⅆ                  n                                                  ⅆ                                      t                    n                                                              ⁡                              [                                                                            ⅆ                                              -                        m                                                              ⁢                                          f                      ⁡                                              (                        t                        )                                                                                                  ⅆ                                          t                                              -                        m                                                                                            ]                                      =                                          1                                  Γ                  ⁡                                      (                    m                    )                                                              ⁢                                                ⅆ                  n                                                  ⅆ                                      t                    n                                                              ⁢                                                ∫                  0                  t                                ⁢                                                                            (                                              t                        -                        τ                                            )                                                              m                      -                      1                                                        ⁢                                      f                    ⁡                                          (                      τ                      )                                                        ⁢                                      ⅆ                    τ                                                                                                          (        4        )            
In view of the foregoing background, an object of the present invention is to provide a circuit arrangement adapted to implement non-integer order dynamic systems, such as, e.g., non-integer order PID controllers.
This and other objects, advantages and features in accordance with the present invention are provided by a circuit for implementing a non-integer order dynamic system, wherein the circuit comprises a neural network for receiving at least one input signal and for generating therefrom at least one output signal. The at least one input and output signals are preferably related to each other by a non-integer order integro-differential relationship through coefficients of the neural network.
The circuit may further include at least one input layer register for receiving the input signal, and at least one neuron comprising at least one adder block for calculating an input thereof based on the at least one input signal and a respective set of coefficients. The at least one neuron may comprise at least one weighing function block for computing an output thereof based upon a respective weighing function. The weighing function may be based upon a sigmoidal function. At least one output layer adder circuit may be used for calculating the at least one output of the neural network from outputs of the at least one neuron.
The at least one neuron may comprise a plurality of neurons, and the circuit further comprises a multiplexer for applying respective weighing functions to the plurality of neurons.
Another embodiment of the present invention is directed to a non-integer order proportional, integral and differential (PID) controller comprising a neural network for receiving at least one input signal and for generating therefrom at least one output signal. The at least one input and output signals are preferably related to each other by a non-integer order integro-differential relationship through coefficients of the neural network.
The PID controller further comprises a proportional block for adding to the at least one output signal an input signal component subject to a proportional weighing coefficient, an integral block for adding to the at least one output signal a non-integer integral signal component subject to an integral weighing coefficient, and a derivative block for adding to the at least one output signal a non-integer derivative signal component subject to a derivative weighing coefficient.
A first input register may be associated with the integral block for feeding an output of the integral block back to an input of the neural network, and a second input register may be associated with the derivative block for feeding an output of the derivative block back to the input of the neural network.
Yet another embodiment of the present invention is directed to a system comprising a plurality of non-integer order PID controllers connected together as defined above. In particular, at least one of the integral and derivative blocks in one of the plurality of PID controllers generates at least one respective output signal which is fed to at least one of the integral and derivative blocks in another PID controller.
Another aspect of the present invention is directed to a method for implementing a non-integer order dynamic system. The method preferably comprises receiving at least one input signal using a neural network and for generating therefrom at least one output signal, the at least one input and output signal being related to each other by a non-integer order integro-differential relationship through coefficients of the neural network.